Another case of importance is the Discrete Time Fourier Transform
(DTFT), which is like the DFT except that the transform accepts an
infinite number of samples instead of only . In this case,
frequency is continuous, and

The DTFT is what you get in the limit as the number of samples in the
DFT approaches infinity. The lower limit of summation remains zero
because we are assuming all signals are zero for negative time (such
signals are said to be causal). This means we are working with
unilateral Fourier transforms. There are also corresponding
bilateral transforms for which the lower summation limit is
. The DTFT is discussed further in
§B.1.

and this is the definition of the transform. It is a
generalization of the DTFT: The DTFT equals the transform evaluated on
the unit circle in the plane. In principle, the transform
can also be recovered from the DTFT by means of ``analytic continuation''
from the unit circle to the entire plane (subject to mathematical
disclaimers which are unnecessary in practical applications since they are
always finite).

Why have a transform when it seems to contain no more information than
the DTFT? It is useful to generalize from the unit circle (where the DFT
and DTFT live) to the entire complex plane (the transform's domain) for
a number of reasons. First, it allows transformation of growing
functions of time such as growing exponentials; the only limitation on
growth is that it cannot be faster than exponential. Secondly, the
transform has a deeper algebraic structure over the complex plane as a
whole than it does only over the unit circle. For example, the
transform of any finite signal is simply a polynomial in . As
such, it can be fully characterized (up to a constant scale factor) by its
zeros in the plane. Similarly, the transform of an
exponential can be characterized to within a scale factor
by a single point in the plane (the
point which generates the exponential); since the transform goes
to infinity at that point, it is called a pole of the transform.
More generally, the transform of any generalized complex sinusoid
is simply a pole located at the point which generates the sinusoid.
Poles and zeros are used extensively in the analysis of recursive
digital filters. On the most general level, every finite-order, linear,
time-invariant, discrete-time system is fully specified (up to a scale
factor) by its poles and zeros in the plane. This topic will be taken
up in detail in Book II [68].

In the continuous-time case, we have the Fourier transform
which projects onto the continuous-time sinusoids defined by
, and the appropriate inner product is

Finally, the Laplace transform is the continuous-time counterpart
of the transform, and it projects signals onto exponentially growing
or decaying complex sinusoids:

The Fourier transform equals the Laplace transform evaluated along the
`` axis'' in the plane, i.e., along the line , for
which . Also, the Laplace transform is obtainable from the
Fourier transform via analytic continuation. The usefulness of the Laplace
transform relative to the Fourier transform is exactly analogous to that of
the transform outlined above.